Chapter 5.5, Problem 57PPS

a.

To determine

Construct a rectangle $ARQD$ .

Answer to Problem 57PPS

  

Explanation of Solution

Given information:

Use quadratic equations to investigate golden rectangles and the golden ratio.

Calculation:

The geometric representaion is shown below:

Draw square

  

Locate the midpoint $\overline{CD}$ label it $P$ , join $\overline{PB}$

Now cut an arc with radius $\overline{PB}$ from $B$ to clockwise till base of the square.

Extend $\overline{CD}$ to the arc meeting point $Q$ .

  

Hence, the rectangle $ARQD$ drawn.

b.

To determine

Solve $\frac{DQ}{AD}=\frac{QR}{CQ}forx$ by completing the square method.

Answer to Problem 57PPS

  $x=\frac{1\pm \sqrt{5}}{2}$

Explanation of Solution

Given information:

Use quadratic equations to investigate golden rectangles and the golden ratio.

  $AD=xandCQ=1$

Calculation:

Let $AD=x\&CQ=1$ , then;

  $QR=1$

Hence,

  $DQ=x+1$

And,

  $\frac{DQ}{AD}=\frac{QR}{CQ}$

Substitute

  $\begin{array}{l}\frac{x+1}{x}=\frac{x}{1}\\ x+1=x^2\\ x^2-x-1=0\end{array}$

The value $x$ is determined as follows:

  $\begin{array}{l}x=\frac{1\pm \sqrt{1+4}}{2}\\ x=\frac{1\pm \sqrt{5}}{2}\end{array}$

The value $x$ is not negative, hence;

  $x=\frac{1\pm \sqrt{5}}{2}$

c.

To determine

Make a table for $xandCQ$

Answer to Problem 57PPS

  $\begin{array}{cc}CQ& x\\ 2& 1+\sqrt{5}\\ 3& \frac{3+3\sqrt{5}}{2}\\ 4& 2+2\sqrt{5}\end{array}$

Explanation of Solution

Given information:

Use quadratic equations to investigate golden rectangles and the golden ratio.

  $CQ=2,3,4$

Calculation:

The table is shown below;

  $\begin{array}{cc}CQ& x\\ 2& 1+\sqrt{5}\\ 3& \frac{3+3\sqrt{5}}{2}\\ 4& 2+2\sqrt{5}\end{array}$

d.

To determine

Write the equation to find $x$ .

Answer to Problem 57PPS

  $x=\frac{n\left(1+\sqrt{5}\right)}{2}$

Explanation of Solution

Given information:

Use quadratic equations to investigate golden rectangles and the golden ratio.

  $CQ=n$ where $n$ is non zero real number.

Calculation:

The value of $x$ are multiplies of $\frac{1+\sqrt{5}}{2}$ , hence:

  $x=\frac{n\left(1+\sqrt{5}\right)}{2}$